Talk:Homotopy groups of spheres/Archive 2

Archive 1

Archive 2

Balance

I think the article is unbalanced insofar as it contains plenty of tables whose content I consider less encyclopaedic. Seriously, isn't the value of pi_30(S_20) something we could refer the reader to a (printed or online) reference or to an appropriate subarticle? Unless we do a more thorough discussion of the values, I feel it is not that much sense in merely listing the values. For example, I like the table of stable h.g.- everybody will notice the kind of periodicity in the complexity of the groups. But the subsection fails a little bit to explain these. I don't know if it is somehow (perhaps vaguely) possible to give indications on this. Is it? If so, it has to be there.

Another thing: a while ago I added a sentence relating the existence of Hopf fibrations to the theory of division algebras over R. KSmrq deleted this calling it a "cleanup". I don't know/understand why he did so. I still think this kind of remark is actually more worthy than listing values whose principal effect is to convey to the reader a feeling of helplessness.

Another way to tailor down the amount of table values: the three tables contain a considerable overlap. I know there are arguments for either presentation, but altogether the article contains more letters than spirit, so to say. Jakob.scholbach 12:59, 26 October 2007 (UTC)

As the diff shows, my edit was a cleanup, of a number of scattered items. I decided to delete the sentence in question

• "The occurrence of these fibrations and the fact that there are no similar higher-dimensional analoga is explained by the theory of division algebras over the real numbers, thus exhibiting a link between topology and abstract algebra and number theory."

because it struck me as an unhelpful digression. (It drew my attention because of "analoga", presumably a typo.) In fact, "is explained" overstates the case, since division algebras are just one possible approach.

The content of the article will be what we make it. At the moment, "we" is mostly R.e.b. (talk · contribs) adding substance, with a little polishing by Geometry guy (talk · contribs) and me. I did write a first draft of a more substantial Definitions section, but when I asked for comments only Geometry guy responded. I expect to rework the Definitions, and perhaps flesh out the discussion of lower examples a little. The major idea I would like to see amplified is how we use sequences, exact or spectral, to compute groups. Although I did put a little in concerning the fibration sequences (as we discussed not long ago), we could really use a section with a more thorough discussion of the process. Also, if you want to work on polishing and structure, I believe R.e.b. is working mostly from Hatcher's online books, so you could read up and add a little more kindness for our readers. --KSmrq^T 19:23, 26 October 2007 (UTC)

I believe the tables are encyclopedic, especially as Wikipedia's goal is to contain the "sum of all human knowledge" (subject, of course, to WP:N and WP:NOT). So the real question is, should they be in this article, or are they unnecessary detail, which could be filled in by an article on Tables of homotopy groups of spheres? My feeling is that they should stay, especially as they are much more appealing that the technically advanced paragraphs. I agree though, that they should be accompanied by more explanation, and I think R.e.b. has been adding some.

As for the division algebras sentence, I would have just re-added it, maybe rephrased. If KSmrq really thinks it is a big deal (which I doubt), he would have removed it again with a more specific edit summary... (In general, reverting a revert is not impolite, IMO, but after the second revert it is time to talk!) Geometry guy 20:01, 26 October 2007 (UTC)

OK, I tried to not only criticise, but also give some (more) constructive input. The results of this nocturnal effort are here (2nd section). I thoroughly read your, KSmrq, proposal, which I like insofar as it is accessible, but IMO brings too much notions which are not necessary to spell out that explicitly (homeomorphic, manifold, equivalence classes, associativity, suspension). I tried to give an introduction as layman-friendly as I could come up with. Compared to KSmrq's proposal, more emphasis is put on the general mathematical concept of category (this is an everyday word, fortunately :-)) and the impact of a functor (without telling that we have a functor). I (would) have some more images illustrating the story. Most of this is pretty intuitive, so I think images can convey almost all of the content. In addition to such an "Introduction" section, I would write a more formal and also much shorter "definitions" section.

As for the "cleanup": I will try to find a reference backing up my statement and if I find it, restore the statement. Saying why there is a fibration for certain indices equal to 3, 7 and 15 is IMO not at all unhelpful, but there is certainly a neater way to put it than I did. (For my "analoga": we say so in German, being somewhat closer to our ancient greek roots, it seems...) Jakob.scholbach 23:22, 26 October 2007 (UTC)

A few people have asked for more images, anything in particular? --Salix alba (talk) 23:56, 26 October 2007 (UTC)

Well, if you could do that, it would be great. I'm personally not at all into animated gifs, so I would have some trouble doing this. I hope the descriptions of what I had in mind (see here) are clear enough. Thanks. Jakob.scholbach 10:55, 28 October 2007 (UTC)

I agree that it is important that we keep the tables. In a subarticle, maybe, but they are very important in homotopy theory, and represent vast amounts of human research. I would personally keep tables as large as known on the homotopy groups of spheres, Lie groups, Grassmannian manifolds, Stiefel manifolds, flag manifolds and so forth, but this may be POV. An encyclopedia is a reference work, and the homotopy groups of spheres are the periodic table of the homotopy theory world. For what it's worth, I have a PhD in topology. Adam1729 (talk) 20:48, 23 December 2007 (UTC)

I concur — in fact, I'll see you, and raise you one. The tables as they are only include part of the story — the additive structure of π_*(Sⁿ), but not the "multiplicative" structure, and other, higher-order operations. This is mentioned (too briefly, in my opinion), in the Ring structure subsection, but not fully developed in the tables, so as to make it clear(er) what it means, and how useful that is. To make them "come alive", and reveal the deeper meaning behind those numbers, one must pay attention to those operations, I think. So, instead of wringing our hands about whether to keep or not to keep those tables, POV vs NPOV (what's that got to do with this, as close to pure knowledge as one is likely to get to?), and so on, how about working on having even more spiffy tables? Cheers, — Turgidson (talk) 22:09, 23 December 2007 (UTC)

Algorithm to compute homotopy groups of spheres

Is there an algorithm to compute all the homotopy groups of a sphere? The article mentions the Cartan--Serre "killing homotopy groups" method, but is vague on whether this is a complete method for doing so, albeit unwieldy. The Math Review calls it a "partial method" where one needs additional info to make the computations work. It's unclear to me whether the additional info (about "Eilenberg groups") is known for the case of spheres. In any case, if there is such a algorithm (even if useless in practice), it should be made clearer. --C S (talk) 04:13, 17 November 2007 (UTC)

There is now an algorithm — at least in principle — to compute the homotopy groups of S^2 (and, thereby, of S^3), using suitable braid groups. For higher-dimensional spheres, one still needs spectral sequences and the like, far as I know. Turgidson (talk) 06:29, 17 November 2007 (UTC)

So the question becomes: can the (e.g. Serre) spectral sequence be made computationally explicit? And the answer may actually be "yes": that's what Computational Algebraic Topology packages like Kenzo claim to achieve. Well, I have never took the time to check the precise extent of what Kenzo claims to be able to compute, but I think this includes arbitrary homotopy groups of spheres (at least in principle).

Anyway, I just stumbled on this page and was really impressed by its quality. After reading some of the comments above, I can see why it might indeed use a few further adjustments in structure here and there, but I'll still take my hat off to you guys. Great work! Bikasuishin (talk) 00:10, 24 January 2008 (UTC)

Minor notation fix

S³ is a four dimensional sphere not three as was stated. Made a minor correction on 08/21/2008. —Preceding unsigned comment added by 147.153.247.34 (talk) 02:36, 22 August 2008 (UTC)

Um no, the dimension refers to its dimension as a manifold not the ambient space it lives in, hence the normal sphere in 3 space, S², is two dimensional not three. --Salix alba (talk) 05:06, 22 August 2008 (UTC)

Table

I'm not completely convinced by some of R.e.b.'s recent changes to the table. In particular, I think listing the homotopy groups of S⁰ is not such a good idea, as the zero sphere is not connected, so we would have to start discussing base-point issues. Also, I find the stabilization along the diagonal less striking now that all the entries are filled in. Views? Geometry guy 15:48, 10 October 2007 (UTC)

Also, why the change of the trivial group notation to 1 instead of 0 which is inconsistent with the rest of the article?--agr 17:02, 10 October 2007 (UTC)

We should be consistent across the article. As a related issue, we should decide whether we want to use multiplicative or additive notation for abelian groups. Essentially R.e.b. has changed the tables to multiplicative notation, although Z is still used for the infinite cyclic group, which is really an additive notation. Geometry guy 17:49, 10 October 2007 (UTC)

If you are using m as an abbreviation for the cyclic group of order m, then the trivial group is 1, not 0, as it has 1 element. Moreover one should then use a.b rather than a+b for the sum of the two groups, as this gives the correct order. I agree there is a good case for changing Z to ∞ but I couldnt be bothered to do this, since Z seemed harmless. R.e.b. 22:11, 10 October 2007 (UTC)

About S⁰: the article defines homotopy groups incorrectly. The way to fix this is not to eliminate mention of the perfectly respectable sphere S⁰, but to fix the definition. R.e.b. 22:37, 10 October 2007 (UTC)

The homotopy classes do not form a group under composition, unlike for For S^n>0. Tompw (talk) (review) 11:56, 13 October 2007 (UTC)

I've been thinking about the table, and I made a version that I like on a user subpage. It has the following features:

• Standard notation (no need for an explanation at the beginning)
• Simple layout, with one column for each π_n and one row for each sphere
• Colors to show the stable homotopy groups, with a dark line showing the stable/unstable barrier
• Serre's infinite groups are highlighted in red.

The disadvantage is that it contains less information than the present table, although I'm not sure that such an extensive list is necessary for a Wikipedia article. Possibly it could be extended to include more data. Jim 07:01, 11 October 2007 (UTC)

I like the layout of Jim's table much better than what's presently in the article. And I also don't think that it's necessary to reproduce here everything that is known about the homotopy groups of spheres. Arcfrk 07:37, 11 October 2007 (UTC)

The notation for abelian groups in the new table is much better, and putting spheres in rows is also an improvement; these were problems I inherited from the original version. The main problem is that the new table discards most of the information, which is a bad idea as it no longer goes far enough to show many of the more interesting phenomena. There are also good reasons why almost every published table arranges by π_n+k rather then by π_n: this makes several subtle pattern much easier to see. R.e.b. 15:20, 11 October 2007 (UTC)

The two tables have complementary advantages, and the article could use both: Jim's table is easier for beginners and could go in the section on low dimensional homotopy groups, while the large compressed table could stay at the end for more serious readers. R.e.b. 16:16, 11 October 2007 (UTC)

I've added it to the beginning of the "general theory" section, with some explanation of the coloring immediately below. Jim 23:24, 11 October 2007 (UTC)

The table is nice, however the colors are really hard to distinguish. Perhaps something less bland? KSmrq has at his user page a couple of colors which are very well contrasting. Jakob.scholbach 21:00, 12 October 2007 (UTC)

Strong colours can't be used as background colours for accessibility reasons. Already some of these colours may be too dark for the visually impaired. (Continued below.) Geometry guy 22:25, 12 October 2007 (UTC)

OK. Well, I'm not visually impaired, but nonetheless (with a laptop screen) the colors are distinguishable only in a small angular range. Perhaps one could try hatching the cells in a different way. Or somehow color the border of every cell (this could be done with a strong color). Jakob.scholbach 07:49, 13 October 2007 (UTC)

On my cheap (and common) flat screen, the paler blue is very hard to tell form the blue. I've also increased the cellpadding because IE wasn't taking accoutn of the subscripts in the left-hand column. Tompw (talk) (review) 11:56, 13 October 2007 (UTC)

Could you find some other way to fix the problem? The cell padding makes the table too wide to fit on standard browser screens.R.e.b. 13:14, 13 October 2007 (UTC)

In the table I made, I fixed the subscript problem by setting the height of each row manually (using the first cell in each row). I also found in necessary to use valign="top" for cells containing superscripts for them to display correctly in IE. Jim 17:36, 13 October 2007 (UTC)

I have tried modifying the colour scheme to make the contrast (red/blue) between the unstable and stable groups more striking. The contrast between even and odd rows is now very little, but I think that is less important. How does it look on a cheap flat screen? Geometry guy 15:10, 14 October 2007 (UTC)

(Continued from above.) I don't like the new tables. I think there should be only one table, and I think it should combine the merits of the two tables that there are now. In particular, the period three colouration has no meaning: periods 2,4 and 8, are relevant, but not period 3. Geometry guy 22:25, 12 October 2007 (UTC)

I'm worried about the period 3 myself. I used it because it is the smallest period that highlights the diagonals effectively. Possibly it would be better if each of the diagonals were a different color?

In any case, I'm not too attached to any particular coloring scheme. But I really think that the basic table of homotopy groups in this article should be arranged with π_k(Sⁿ) in position k,n, and should use standard notation for groups in the table cells. Jim 17:36, 13 October 2007 (UTC)

Period 4 would work for me (since it brings out the 3 mod 4 issue). I think we only need one of these two tables, plus a separate, much simpler, table of the stable groups. I have mixed feelings about which approach I prefer. I like your table, but I see the benefits of the n+k approach, and we really should follow the literature. If you have a source for the (k,n) version, that might swing the balance. Geometry guy 18:06, 13 October 2007 (UTC)

The (k, n) version is used in Hatcher's Algebraic Topology book (Chapter 4, pg. 339). Jim 19:10, 14 October 2007 (UTC)

I've modified the first table with more satuated colours, using Web-safe color. Hopefully sufficient contrast for disabled and between colours. I had to fiddle with my monitor to actually see the previous colours. The period is still three, which helps keep same level of satuation. --Salix alba (talk) 15:44, 14 October 2007 (UTC)

@Geometryguy: on my cheap screen the table looks royal :-). Much better than before. Jakob.scholbach 17:44, 14 October 2007 (UTC)

Where to jump in? I'm curious to see if I can come up with a single complete table; I'll experiment privately.

• Meanwhile, using the wikEd tool and a little hand cleanup I've been able to convert the original HTML table markup to wiki pipe markup. Like Jim's table, it's much easier to deal with!
• Using 1 instead of 0 is inconsistent with the article itself, and with most literature. Better still, leave the trivial group cells blank; we have no missing data so the interpretation would be unambiguous.
• One way to address Jim's preference for "standard notation", meaning Z×Z₂³ instead of ∞.2³ or Z+2³, might be to make a conceptual shift. Can the table list group orders with p-primary factorization, rather than the groups themselves?
• With so many patterns connecting table entries, it's tough to imagine any one view that can visually highlight them all. For example,
  1. Using Jim's π_i(Sⁿ) table, with row n and column i, the equality of the S² and S³ groups jumps out.
  3. Using row n and column k, group n+k, the stable groups are more salient.
  5. Even and odd patterns are important; they take different forms in the two layouts.
  7. Other patterns, such as π_i(S²ⁿ) = π_i−1(S²ⁿ⁻¹)×π_i(S⁴ⁿ⁻¹) modulo 2-torsion, are probably beyond visual cues.
• The maximum table width should not require horizontal scrolling on an 800×600 screen. (See W3C statistics.)
• Ooh, cotton candy colors! The new background colors of the big table might be too dark for good readability contrast. Web-safe colors were never all that safe, especially not for those using 15-bit color depth; and they were chosen with CRTs in mind, whereas LCDs have become increasingly common. This tool is one way to get a feel for how color combinations look, including simulation of various forms of color blindness. A good rule of thumb is to use color as a secondary cue; for example, traffic light standards try to accomodate the 5% of men who have difficulty distinguishing red from green. (Vischeck is a fun way to experiment.)

It might be helpful to compose a wish-list of facts we'd like to indicate in the table, so we can choose the ones that we feel are most important and most practical to show. --KSmrq^T 06:36, 15 October 2007 (UTC)

PS: Two brief remarks about table notation:

• Although many people working on homotopy groups would be comfortable with Z₂, the trend seems to be towards Z/2Z or Z/(2) or even Z/2. For example, Toda (in an older publication) uses Z₂ while Ravenel (much more recently) uses Z/(2). The reason, of course, is to avoid confusion with p-adic numbers.
• The fundamental theorem of finite abelian groups tells us that the group for n = 6, k = 8 that Toda describes as Z₂₄+Z₂, having order 48, can be expressed as the direct sum of Z₃ and groups of the form Z_2^m. Since the same applies to Z₂₄ itself, it is not necessarily a contradiction for another author to list this group as Z₆+Z₈. (See Consistent notation above.) But we do care. For example, Ravenel (after Toda) lists (n = 13,k = 19) as 264.2³, and lists (n = 14,k = 19) as 264.4.2. I therefore assume these are different groups of the same order. Please note that the Jie Wu table is not a direct copy of Toda in notation; he lists (6,8) as 8+2+3, (13,19) as 8+2³+3+11, and (14,19) as 8+4+2+3+11.

Our choice of notation can affect visibility of patterns, as well as accuracy. We currently have a footnote that says "There are a few differences between Ravenal's tables and Toda's tables"; is this really true, regarding substance? (And note that "Ravenal" is a misspelling.) --KSmrq^T 11:19, 15 October 2007 (UTC)

Important note: Z₄ is not the same as Z₂ X Z₂; thus 264.4.2, 264.8, 132.16, 66.32 and 33.64 all different groups. (The fundamental theorem of finite abelian groups tells us that a finite Abelian group of order n can be written in the form $Z_{p_{1}}^{q_{1}}\times Z_{p_{2}}^{q_{2}}\times Z_{p_{3}}^{q_{3}}...$, where $n={p_{1}}^{q_{1}}{p_{2}}^{q_{2}}{p_{3}}^{q_{3}}...$ and p_i are primes which may be repeated.) Tompw (talk) (review) 19:01, 22 August 2008 (UTC)

I think I can live with two tables now, as long as the first one is as short as possible. I've adjusted the colours (inspired by KSmrq's palette). I'm still unhappy about the period 3, but I hope that otherwise, this works for everyone. Geometry guy 23:04, 21 October 2007 (UTC)

Toda's Table

I deleted two sentences:

"See also Appendix 3 of Ravenel (2003). There are a few differences between Ravenel's tables and Toda's tables; it is unclear if these are misprints or corrections.)"

Here's why:

One of the entries in Ravenel's table in his book was \pi_17 S^7 = \Z/24 + Z/4. On Oct. 25, 2009, I emailed Ravenel and said that this contradicted the EHP sequence and also disagreed with Toda's table. He emailed back and said (1) that I was right, (2) he would correct the online version of his book, and (3) the intention when he wrote the book was to copy Toda's table; he did not attempt corrections.

So at the very least the sentence "There are a few differences between Ravenel's tables and Toda's tables; it is unclear if these are misprints or corrections." needs to be deleted. I have not checked whether the online version of Ravenel's book matches Toda's table now. Jfdavis (talk) 19:57, 7 November 2009 (UTC)

No pattern?

Under the "General theory" section, can phrasing like

is often complicated and the results are surprisingly complex, with no pattern discerned to date.

be modified to be more accurate? Surely there are patterns- the next paragraph gives two different patterns. I think what is meant is something like "there is no known algorithm for computing π_n(S^m) for all m and n", though that's perhaps a bit confusing to a layperson (though I'm sure they wouldn't have gotten this far), and has entirely a different feel to it. Staecker 16:12, 8 October 2007 (UTC)

Good point. I suggest you be bold! Geometry guy 21:46, 8 October 2007 (UTC)

I couldn't really think of a rewrite that I like... and I still can't... Staecker 22:27, 8 October 2007 (UTC)

In general, if you want to point out an untrue statement in an article, it's always a good idea to say what the allegedly untrue statement actually is.

I can't imagine how anyone could form an opinion about whether something "is often complicated and the results are surprisingly complex, with no pattern discerned to date" if you keep secret what that something is. Plus, the statement is by no means absolute, using the word "often", not "always".

Of course, I could look at the article and search for the quoted text to find out for myself how that sentence begins. But I have to choose how to spend my time, and it seems a waste of time to try to figure out what someone means who won't even make the effort to say what they mean.Daqu (talk) 05:42, 26 November 2009 (UTC)

Much of this "good" article is incomprehensible

The article has lovely pictures, and if you don't read it carefully, it may appear to be well written.

It is not.

Consider as a typical example this sentence [bracketed phrases are mine]:

"The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S^3 around the usual sphere S^2 in a non-trivial fashion, and so is not equivalent to a one-point mapping."

Let's see now, S^3 is wrapped around S^2 "in a non-trivial fashion" [whatever that means], and "so" is "not equivalent [which in English is called not homotopic]" to a "one-point mapping [which in English is called a constant mapping]".

Well, that certainly explains things. Not.Daqu (talk) 05:54, 26 November 2009 (UTC)

Not a sentence

For:

"Most of the groups are finite. The only unstable groups which are not are either on the main diagonal or immediately above the jagged line (highlighted in yellow)."

The second sentence isn't a complete sentence as far I as I can tell. It looks to me like <blah1> which are not <blah2> but doesn't actually say anything about them.

Additionally I don't see anything yellow. If there is yellow it is too close to white for me to tell on my computer. —Preceding unsigned comment added by 69.33.111.74 (talk) 19:41, 15 February 2010 (UTC) #

Read as The only unstable groups which are not [finite] and it does make good sense. Charles Matthews (talk) 21:42, 15 February 2010 (UTC)

Needs to be fixed

Under "applications", note that the Kervaire invariant question has been resolved except for dimension 126. —Preceding unsigned comment added by 156.56.144.19 (talk) 00:12, 20 May 2010 (UTC)

Simple Z_2 example?

The article goes through simple examples giving trivial and Z groups, but can someone lead us through a simple example giving a finite group? --JWB (talk) 20:17, 4 January 2013 (UTC)

GA status

Criteria 2b asks that the article "provides in-line citations from reliable sources for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines" (bolding added by me). There are no inline citations in this article and some of the statements have been challenged for over five years now. I am hoping someone watching this can provide inline citations (not just to the statements marked but to all the statements that need them). Otherwise I am afraid that this article will have to undergo a reassessment. AIRcorn (talk) 23:07, 15 March 2013 (UTC)

@Aircorn: Currently, I see 2 cn tags. If the two challenged statements were to be removed, would the article still meet the criteria? The article has no quotations, statistics, or opinions, so challenged material is currently the only issue.--FutureTrillionaire (talk) 23:33, 29 September 2013 (UTC)

I have added the requested references and removed the template at the top of the article. Jakob.scholbach (talk) 09:34, 30 September 2013 (UTC)

Cool. I guess the problem is solved then.--FutureTrillionaire (talk) 15:37, 30 September 2013 (UTC)

Lead image

While the keyring model will always hold a special place in my heart and in the history of this article, I think it is not as illustrative as other newer pictures. On the Hopf fibration page, it has been replaced with one that I made. There are other good pictures of the Hopf fibration online, and this article itself now has other pictures which more effectively communicate what it means to "wrap" one sphere around another. Let's not keep the keyring picture merely because we are used to it and like its historical significance. Is there an expository reason not to replace it?

For reference, here are the two pictures I'm referring to

• 
  The Hopf fibration is a nontrivial mapping of the 3-sphere to the 2-sphere.
• 
  Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.

— Preceding unsigned comment added by Nilesj (talk • contribs) 22:05, 25 February 2019 (UTC)

The comments above are from February 2019, and there hasn't been any further discussion. I think I'll go ahead and make the change. Nilesj (talk) 14:39, 18 July 2019 (UTC)

I missed the original comment, but I think changing it would be fine. –Deacon Vorbis (carbon • videos) 14:43, 18 July 2019 (UTC)

Stable homotopy group

It seems that oeis:A048648 gives the orders of the stable homotopy groups that differ from those given in the article for k = 23, 29..33. It seems that the OEIS numbers originate from the program referenced here, which ultimately references Ravenel's book, just like this article. So the question arises: which data is correct? --colt_browning (talk) 14:27, 4 March 2020 (UTC)

Never mind, the OEIS is now corrected. --colt_browning (talk) 07:59, 25 August 2020 (UTC)

Reminiscent of "The Wizard of Oz" movie

The way the table of π_n(S^k) changes to three different colors all of a sudden in the stable range is a) delightful and b) reminiscent of how the movie "The Wizard of Oz" morphs from black & white to color, once the protagonist has entered the Land of Oz.

However: Has anyone proven that three colors suffice for coloring the stable range of π_nS^k ??? — Preceding unsigned comment added by 2601:200:C000:1A0:DD44:41FE:AA4:A892 (talk) 23:19, 28 November 2022 (UTC)